After exploring uncoupled, partially coupled, and fully coupled systems, we now turn to the most tractable and widely used class: linear systems with constant coefficients.
Their algebraic form is simple enough for exact analysis, yet flexible enough to model everything from harmonic oscillators to interacting populations. Our first task is to define what “linear” means and show how a short piece of matrix notation can replace a page-full of equations.
A system is linear if every equation is linear with respect to its dependent variables (all to the power of one). For two variables \(x(t)\) and \(y(t)\) the general form is
\begin{align*}
\frac{dx}{dt} \amp = a x + b y\\
\frac{dy}{dt} \amp = c x + d y
\end{align*}
where \(a\text{,}\)\(b\text{,}\)\(c\text{,}\) and \(d\) are constants. The system is therefore autonomous (no explicit \(t\)) and planar (two-dimensional).
\begin{align*}
\frac{dx}{dt} \amp = a x + b y\\
\frac{dy}{dt} \amp = c x + d y
\end{align*}
define the coefficient matrix and state vector
\begin{equation*}
A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix},
\qquad
Y = \begin{bmatrix} x \\ y \end{bmatrix},
\qquad
\frac{dY}{dt} = \begin{bmatrix} \dfrac{dx}{dt} \\ \dfrac{dy}{dt} \end{bmatrix}
\end{equation*}
This compact format presents the system as a single equation, linking directly to linear-algebra tools such as eigenvalues, which we will explore later.
The dimension of the system equals the length of \(Y\text{.}\) Our focus in this chapter remains on the planar case (\(n=2\)) where geometry and algebra meet most clearly.
